density

created by Woundweavr
(idea) by Woundweavr (4.1 y) (print)   (I like it!) Sat Nov 13 1999 at 14:30:30
The amount of matter, or the mass of an object, divided by the volume it is contained in.
Also the amount of people in a given area. See population density.
(idea) by arfarf (1.6 y) (print)   (I like it!) Sat Nov 13 1999 at 14:30:30
If you hold two objects with the same volume, the one that is heavier has the higher density.

Objects will float in liquids with a higher density. The volume of liquid they displace weighs more than the equal, displacing volume. Eureka!

(idea) by ariels (3 d) (print)   (I like it!) 2 C!s Wed Sep 25 2002 at 11:44:14

Half the integers are even. We all know that, but what does that mean? We say that "half the numbers in (0,1) are less than 1/2", because m((0,1/2))=1/2=1/2*m((0,1)), where m(I) is the length or, more generally, the Lebesgue measure of I. But there is no uniform distribution on the integers! So we cannot hope to find some measure μ for which μ(even numbers)=1/2*μ(integers).

Density is an alternative attempt to formalise the statement at the top. For any (large) n, we can say how many integers 1,...,n are even: either n/2 or (n-1)/2. So either 1/2 or 1/2-1/(2n) out of 1,...,n are even. And the limit of this expression for large n is 1/2. We say that the density of the even numbers in the natural numbers is 1/2.

Definition

Let A={a1,...} be a countable set. Typically A=N={1,2,...}. But note that the particular enumeration chosen influences the concept of density very strongly! Let X⊆A be some subset. The density of X in A is

limn→∞ |X∩{a1,...,an}| / n
(if the limit exists; otherwise, we say X has no density).

Replacing "lim" with "lim sup" or "lim inf", we get "upper density" and "lower density", respectively; note that these always exist, and are between 0 and 1.

Easy facts

  • X has density iff its lower and upper densities are equal.
  • The density of X is ≥ 0 and ≤ 1.
  • If X is finite, its density is 0. If we add or remove finitely many elements of X, its density is unchanged.
  • Density is finitely additive: if X and Y are disjoint sets which both have density, then X∪Y has density, which is the sum of the densities of X and Y.
  • Unlike measure, density is not sigma additive.
  • By re-ordering A, we can get any density in [0,1] for any infinite subset X.

Examples

  1. The set of even integers has density 1/2.
  2. The sets of squares, primes, perfect numbers (and others) have density 0.
  3. Not every set has density. For instance,
    X = {1,3,4,9,10,11,12,13,14,15,16,...} = {1} ∪ {21+1,...,22} ∪ ... ∪ {22k-1+1,...,22k} ∪ ...
    has no density (its lower density is 1/2, as seen by taking n=22k-1, while its upper density is 3/4, as seen by taking n=22k).

One can view density as a special case of Césaro means, or even summability, of the characteristic function of X.

(definition) by Webster 1913 (print) Tue Dec 21 1999 at 22:59:26

Den"si*ty (?), n. [L. densitas; cf. F. densit'e.]

1.

The quality of being dense, close, or thick; compactness; -- opposed to rarity.

2. Physics

The ratio of mass, or quantity of matter, to bulk or volume, esp. as compared with the mass and volume of a portion of some substance used as a standard.

⇒ For gases the standard substance is hydrogen, at a temperature of 0° Centigrade and a pressure of 760 millimeters. For liquids and solids the standard is water at a temperature of 4° Centigrade. The density of solids and liquids is usually called specific gravity, and the same is true of gases when referred to air as a standard.

3. Photog.

Depth of shade.

Abney.

 

© Webster 1913.

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